English

The Maximum Number of Appearances of a Word in a Grid

Combinatorics 2022-07-26 v1

Abstract

How can you fill a 3×33\times 3 grid with the letters A and M so that the word ``AMM'' appears as many times as possible in the grid? More generally, given a word ww of length nn, how can you fill an n×nn\times n grid so that ww appears as many times as possible? We solve this problem exactly for several families of words, and we asymptotically solve this problem in higher-dimensional grids.

Keywords

Cite

@article{arxiv.2207.11273,
  title  = {The Maximum Number of Appearances of a Word in a Grid},
  author = {Gregory Patchell and Sam Spiro},
  journal= {arXiv preprint arXiv:2207.11273},
  year   = {2022}
}

Comments

21 pages, American Mathematically Monthly